
H. Cevikalp, D. Larlus, M. Neamtu, B. Triggs, and F. Jurie, Manifoldbased local
classifiers: Linear and nonlinear approaches, J. Signal Processing Systems, to appear.
Abstract:
The Klocal hyperplane distance nearest neighbor (HKNN) algorithm is a local classification
method that builds nonlinear decision surfaces by using locally linear manifolds directly in the original
sample space. Although it has been successfully applied in several classification tasks, it is limited to using
the Euclidean distance metric, which is a significant limitation in the practice. In this paper we reformulate
HKNN in terms of subspaces, and propose a variant, the Local Discriminative Common Vector (LDCV)
method, that is more suitable for classification tasks where the classes have similar intraclass variations.
We then extend both methods to the nonlinear case by using the kernel trick to map the data into a higherdimensional
space, in which the linear manifolds are constructed. This construction allows us to use a wide
variety of distance functions for the local classifiers, while computing distances between the query sample
and the nonlinear manifolds remains straightforward owing to linear nature of the manifolds in the mapped
space. We tested the proposed methods on several classification tasks, obtaining better results than both
the Support Vector Machines (SVMs) and their local counterpart SVMKNN on the USPS and Image
segmentation databases, and outperforming the local SVMKNN on the Caltech and Xerox10 visual
recognition databases.

H. Cevikalp, M. Neamtu, and A. Barkana, The Kernel Common Vector Method: A Novel
Nonlinear Subspace Classifier for Pattern Recognition, IEEE Transactions on Systems, Man, Cybernetics,
B: Cybernetics 37 (2007), 937951.
Abstract:
The common vector (CV) method is a linear subspace classifier method which allows one to discriminate
between classes of data sets, such as those arising in image and word recognition. This method utilizes
subspaces that represent classes during classification. Each subspace is modeled such that common features
of all samples in the corresponding class are extracted. To accomplish this goal, the method eliminates
features that are in the direction of the eigenvectors corresponding to the nonzero eigenvalues of the
covariance matrix of each class. In this paper, we introduce a variation of the CV method, which will
be referred to as the modified CV (MCV) method. Then, a novel approach is proposed to apply the MCV
method in a nonlinearly mapped higher dimensional feature space. In this approach, all samples are
mapped into a higher dimensional feature space using a kernel mapping function, and then, the MCV
method is applied in the mapped space. Under certain conditions, each class gives rise to a unique CV,
and the method guarantees a 100% recognition rate with respect to the training set data. Moreover,
experiments with several test cases also show that the generalization performance of the proposed kernel
method is comparable to the generalization performances of other linear subspace classifier methods as
well as the kernelbased nonlinear subspace method. While both the MCV method and its kernel counterpart
did not outperform the support vector machine (SVM) classifier in most of the reported experiments, the
application of our proposed methods is simpler than that of the multiclass SVM classifier. In addition,
it is not necessary to adjust any parameters in our approach.

M. Neamtu, Delaunay configurations and multivariate splines:
A generalization of a result of B. N. Delaunay,
Trans. Amer. Math. Soc. 359 (2007), 29933004.
Abstract: In the 1920s, B. N. Delaunay
proved that the dual graph of the
Voronoi diagram of a discrete set of points in a Euclidean space
gives rise to a collection of simplices, whose circumspheres
contain no points from this set in their interior. Such Delaunay
simplices
tessellate the convex hull of these points. An
equivalent formulation of this property is that the characteristic
functions of the Delaunay simplices form a partition of unity.
In the paper this result is generalized to the socalled Delaunay
configurations. These are defined by considering all simplices for
which the interiors of their circumspheres contain a fixed number
of points from the given set, in contrast to the Delaunay
simplices, whose circumspheres are empty.
It is proved that every family of Delaunay configurations
generates
a partition of unity, formed by the socalled simplex splines.
These are compactly supported piecewise polynomial functions
which are multivariate analogs of the wellknown univariate
Bsplines. It is also shown that the linear span of the
simplex splines contains all algebraic polynomials of degree not
exceeding the degree of the splines.

H. Cevikalp, M. Neamtu, and M. Wilkes,
Discriminative common vector method with kernels,
IEEE Trans. Neural Networks 17(6), 2006, 15501565.
Abstract:
In some pattern recognition tasks, the dimension of the sample space is larger than the number of the samples in
the training set. This is known as the "small sample size problem". The Linear Discriminant Analysis (LDA)
techniques cannot be applied directly to the small sample size case. The small sample size problem is also
encountered when kernel approaches are used for recognition. In this paper we try to answer the question of "How
should we choose the optimal projection vectors for feature extraction for the small sample size case?" Then, we
propose a new method called the Kernel Discriminative Common Vector (Kernel DCV) method, based on our
findings. In this method, we first nonlinearly map the original input space to an implicit higherdimensional feature
space through a kernel mapping, where the data are hoped to be linearly separable. Then, the optimal projection
vectors are computed in the transformed space. The proposed method yields an optimal solution for maximizing the
modified Fisher's Linear Discriminant criterion given in the paper. Thus, a 100% recognition rate is always
guaranteed for the training set samples. Experiments on test data sets also show that the generalization ability
of the proposed method outperforms other kernel approaches in many situations.

H. Cevikalp, M. Neamtu, and M. Wilkes, Nonlinear discriminative common vector
method, Proceedings of the 9thWorld MultiConference on Systemics, Cybernetics and Informatics,
electronic proceedings, Orlando, 2005.

H. Cevikalp and M. Neamtu, Nonlinear common vectors for pattern classification, In
the 13th European Signal Processing Conference, electronic proceedings, Antalya, Turkey, 2005.

H. Cevikalp, M. Neamtu, and M. Wilkes, Nonlinear discriminant common vectors,
Proceedings IEEE 13th Signal Processing and Communications Applications, Kayseri, Turkey,
pp. 292295, 2005.

B. Dembart, D. Gonsor, and M. Neamtu,
Bivariate quadratic Bsplines used as basis functions for
collocation, in Mathematics for Industry: Challenges and Frontiers 2003.
A Process View: Practice and Theory,
D. R. Ferguson and T. J. Peters (eds.), pp. 178198, Toronto, Ontario, 2005, SIAM.
Abstract:
We present results summarizing the utility of bivariate Bsplines for
solving data fitting and related problems. These basis functions are
defined by certain collections of points in the plane, called knots.
The Bsplines are piecewise quadratic compactlysupported functions,
possessing optimal order of differentiability (C^1). The linear span
of these functions gives rise to a spline space with good approximation
properties. Our experimental results show that the Bspline basis also
entertains excellent spectral properties, rendering the Bsplines
useful for, among other things, iterative solution of data fitting
and scattering problems in computational electromagnetics.

H. Cevikalp, M. Neamtu, M. Wilkes, and A. Barkana,
Discriminative common vectors for face recognition,
IEEE Trans. Pattern Analysis and Machine Intelligence 27 (2005), 413.
Abstract:
In face recognition tasks, the dimension of the sample space is typically
larger than the number of the samples in the training set. As a
consequence, the withinclass scatter matrix is singular and the Linear
Distriminant Analysis (LDA) method cannot be applied directly. This
problem is known as the "small sample size problem". In this paper,
we propose a new face recognition method called the Discriminative
Common Vector method, based on a variation of Fisher's Linear
Discriminant Analysis for the small size case. Two different
algorithms are given to extract the discriminative common vectors
representing each person in the training set while the other uses
the subspace methods and the GramSchmidt orthogonalization
procedure to obtain the discriminative common vectors. The thesese
vectors are used for classification of new faces. The proposed
method yields an optimal solution for maximizing the modified
Fisher's Linear Discriminant criterion given in the paper. Our
results show that the Discriminative Common Vector method is
superior to other methods in terms of recognition accuracy,
efficiency, and numerical stability.

H. Cevikalp, M. Neamtu, M. Wilkes, and A. Barkana,
A novel method for face recognition, Proceedings IEEE 12th Signal Processing
and Communications Applications, 2004, 579582.

M. Neamtu and L. Schumaker,
On the approximation order of splines on spherical triangulations,
Adv. Comput. Math. 21 (2004), 320.
Abstract: Bounds are provided on how well functions
in Sobolev spaces on the sphere can
be approximated by spherical splines, where a spherical spline
of degree $d$ is a $C^r$ function whose pieces are the restrictions
of homogoneous polynomials of degree $d$ to the sphere. The
bounds are expressed in terms of appropriate seminorms
defined with the help of radial projection,
and are obtained using appropriate quasiinterpolation operators.

K. Kopotun, M. Neamtu, and B. Popov,
Weakly nonoscillatory schemes for scalar conservation laws,
Math. Comp. 72 (2003), 17471767.
Abstract:
A new class of Godunovtype numerical methods for solving
nonlinear scalar conservation laws in one space dimension is introduced.
This new class of methods,
called weakly nonoscillatory (WNO), is a generalization of the classical
nonoscillatory schemes. Under certain conditions, convergence and error
estimates for the methods are proved. Examples of such WNO schemes include
modified versions of MinMod and UNO.

T. Morton and M. Neamtu,
Error bounds for solving pseudodifferential equations on spheres by collocation with
zonal kernels,
J. Approx. Theory 114 (2002), 242268.
Abstract:
The problem of solving pseudodifferential equations on spheres
by collocation with zonal kernels is considered and bounds for
the approximation error are established. The bounds are given
in terms of the maximum separation distance of the collocation
points, the order of the pseudodifferential operator, and the
smoothness of the employed zonal kernel. A byproduct of the
results is an improvement on the previously known convergence
order estimates for Lagrange interpolation.

M. Neamtu,
Splines on surfaces,
Handbook on CAGD, NorthHolland, Amsterdam, 2002, 229253.
Abstract:
This chapter addresses the topic of ``splines on surfaces'', an area
of spline theory concerned
with the construction of functions defined on
manifolds in threedimensional Euclidean space. For the most part,
the mathematical aspects of this discipline are in their infancy
and therefore much of what we will say here has an exploratory
character.

D. Gonsor and M. Neamtu,
Can subdivision be useful for geometric modeling applications?,
Boeing Technical Report #01011, 2001, 43pp.
Abstract: The utility of subdivision techniques
is investigated from the point of view of geometric modeling
applications. This report summarizes the findings and
recommendations of the authors concerning the usefulness of
subdivision surfaces for geometric modeling, and in particular
for engineering applications.

M. Neamtu,
What is the natural generalization of univariate splines to higher dimensions?,
in Mathematical Methods for Curves and
Surfaces, T. Lyche and L. L. Schumaker (eds.), Vanderbilt University Press,
Nashville, 2001, pp. 355392.
Abstract:
In the first part of the paper, the problem of defining
multivariate splines in a natural way is formulated and discussed. Then, several
existing constructions of multivariate splines are surveyed, namely
those based on simplex splines. Various difficulties and practical
limitations associated with such constructions are pointed out.
The second part of the paper is concerned with the description of
a new generalization of univariate splines. This
generalization utilizes the novel concept of the socalled Delaunay
configurations, used to select collections of knotsets for simplex
splines. The linear span of the simplex
splines forms a spline space
with several interesting properties. The
space depends uniquely and in a local way on the prescribed knots
and does not require the use of auxiliary or perturbed knots, as is the case
with some earlier constructions. Moreover, the spline space has
a useful structure that makes it possible to represent polynomials
explicitly in terms of simplex splines. This representation closely
resembles a familiar univariate result in which polar forms are used to
express polynomials as linear combinations of the classical Bsplines.

M. Neamtu,
Bivariate simplex Bsplines: A new paradigm,
in Proc. of Spring Conference on Computer Graphics, IEEE Computer
Society, R. Durikovic and S. Czanner (eds.), Los Alamitos, 2001, pp. 7178.
Abstract:A construction of bivariate splines
is described, based on a new concept of higher degree Delaunay
configurations. The crux of this construction is that knotsets
for simplex Bsplines are selected by considering groups of
``nearby'' knots. The new approach
gives rise to a natural generalization of univariate splines
in that the linear span of this collection of Bsplines
forms a space which is analogous to
the classical univariate splines. This new spline space depends
uniquely and in a local way on the prescribed knot locations,
and there is no need to use auxiliary or perturbed knots as in
some earlier constructions.

B. Mulansky and M. Neamtu,
Interpolation and approximation from convex sets II. Infinitedimensional interpolation,
J. Comp. Appl. Math. 119 (2000), 333346.
Abstract:
Let $X$ and $Y$ be topological vector spaces, $A$ be a continuous
linear map from $X$ to $Y$, $C \subset X$, $B$ be
a convex set dense in $C$, and $d \in Y$ be a data point. Conditions
are derived guaranteeing the set $B \cap A^{1}(d)$ to be nonempty and
dense in $C \cap A^{1}(d)$. The paper generalizes earlier results
by the authors to the case where $Y$ is infinite dimensional.
The theory is illustrated with two examples concerning the
existence of smooth monotone extensions of functions defined on
a domain of the Euclidean space to a larger domain.

S. Morigi and M. Neamtu,
Some results for a class of generalized polynomials,
Adv. Comput. Math. 12 (2000), 133149.
Abstract:
A class of generalized polynomials is considered consisting
of the null spaces of certain differential operators with
constant coefficients. This class strictly contains ordinary
polynomials and appropriately scaled trigonometric
polynomials. An analog of the classical Bernstein operator
is introduced and it is shown that generalized Bernstein
polynomials of a continuous function converge to this
function. A convergence result is also proved for degree
elevation of the generalized polynomials. Moreover, the
geometric nature of these functions is discussed and a
connection with certain rational parametric curves is
established.

M. Neamtu,
Convergence of subdivision versus solvability of refinement equations,
East J. Approx. 5 (1999), 183210.
Abstract:
Under the assumption that a given twoscale refinement equation
possesses a continuous solution, called a refinable function,
necessary and sufficient conditions are derived for convergence
of the corresponding univariate stationary subdivision scheme
with a finitely supported mask. These conditions are expressed
using the factorization of the subdivision mask and do not
require the computation of a spectral radius of matrices or
solving an eigenvalue problem.
The main result is that subdivision is convergent if and only if
it is convergent for sequences characterizing linear dependence
relations for integershifts of the refinable function. Moreover,
this function can be generated by employing a convergent
subdivision corresponding to an appropriately chosen mask.
As a consequence of the main results it is shown that subdivision
associated with a nonnegative mask, satisfying a simple condition,
converges if and only if the corresponding refinement equation
possesses a continuous solution.

M. Neamtu, H. Pottmann, and L. L. Schumaker,
Designing NURBS cam profiles using trigonometric splines,
J. Mech. Design 120 (1998), 175180.
Abstract:
We show how to design cam profiles using NURBS curves
whose support functions are appropriately scaled
trigonometric splines. In particular, we discuss the
design of cams with various side conditions of
practical interest, such as interpolation conditions,
constant diameter, minimal acceleration or jerk, and
constant dwells. In contrast to general polynomial
curves, these NURBS curves have the useful property
that their offsets are of the same type, and hence also
have an exact NURBS representation.

M. Neamtu, H. Pottmann, and L. L. Schumaker,
Dual focal splines and rational curves with rational offsets,
Math. Eng. Ind. 7 (1998), 139154.
Abstract:
We review the theory of homogeneous splines and
their relationship to special rational splines
considered by J.~S\'{a}nchezReyes and
independently by P.~de~Casteljau who called them
focal splines. Applying an appropriate duality,
we transform focal splines into a remarkable
class of rational curves with rational offsets.
We investigate geometric properties of these dual
focal splines, and discuss applications to curve
design problems.

B. Mulansky and M. Neamtu,
Interpolation and approximation from convex sets,
J. Approx. Theory 92 (1998), 82100.
Abstract:
Let $X$ be a topological vector space, $Y=\R^n$, $n \in \N$,
$A$ a continuous linear map from $X$ to $Y$, $C \subset X$,
$B$ a convex set dense in $C$, and $d \in Y$ a data point.
We derive conditions which guarantee that the set
$B \cap A^{1}(d)$ is nonempty and dense in $C \cap A^{1}(d)$.
Some applications to shape preserving interpolation and
approximation are described.

M. Neamtu,
Homogeneous simplex splines,
J. Comp. Appl. Math. 73 (1996), 173189.
Abstract:
Homogeneous simplex splines, also known as cone splines
or multivariate truncated power functions, are discussed from
a perspective of homogeneous divided differences and polar
forms. This makes it possible to derive the basic
properties of these splines in a simple and economic way. In
addition, a construction of spaces of homogeneous simplex
splines is considered, which in the nonhomogeneous setting
is due to Dahmen, Micchelli, and Seidel. A proof for this
construction is presented, based on knot insertion. Restricting
the homogeneous splines to a sphere gives rise to spaces
of spherical simplex splines.

P. Alfeld, M. Neamtu, and L. L. Schumaker,
Fitting scattered data on spherelike surfaces using spherical
splines, J. Comp. Appl. Math. 73 (1996), 543.
Abstract:
Spaces of polynomial splines defined on planar triangulations
are very useful tools for fitting scattered data in the plane.
Recently, [\cite{ANS2}, \cite{ANS3}], using homogeneous
polynomials, we have developed analogous spline spaces
defined on triangulations on the sphere and on spherelike
surfaces. Using these spaces, it is possible to construct
analogs of many of the classical interpolation and fitting
methods. Here we examine some of the more interesting ones in
detail. For interpolation, we discuss macroelement methods
and minimal energy splines, and for fitting, we consider
discrete least squares and penalized least squares.

P. Alfeld, M. Neamtu, and L. L. Schumaker,
Dimension and local bases of homogeneous spline spaces,
SIAM J. Math. Anal. 27 (1996), 14821501.
Abstract:
Recently, we have introduced spaces of splines defined on
triangulations lying on the sphere or on spherelike surfaces.
These spaces arose out of a new kind of BernsteinBezier
theory on such surfaces. The purpose of this paper is to
contribute to the development of a constructive theory for
such spline spaces analogous to the wellknown theory of
polynomial splines on planar triangulations. Rather than
working with splines on spherelike surfaces directly, we
instead investigate more general spaces of homogeneous splines
in R^3. In particular, we present formulae for the dimensions
of such spline spaces, and construct locally supported bases
for them.

D. Gonsor and M. Neamtu,
Null spaces of differential operators, polar forms, and
splines, J. Approx. Th. 86 (1996), 81107.
Abstract:
In this article we consider a class of functions, called
$\cD$polynomials, which are contained in the null space of certain
second order differential operators with constant coefficients.
The class of splines generated by these $\cD$polynomials strictly
contains the polynomial, trigonometric and hyperbolic splines.
The main objective of this paper is to present a unified theory
of this class of splines via the concept of a polar form. By
systematically employing polar forms, we extend essentially
all of the wellknown results concerning polynomial splines.
Among other topics, we introduce a Schoenberg operator and
define control curves for these splines. We also examine the
knot insertion and subdivision algorithms and prove that the
subdivision schemes converge quadratically.

P. Alfeld, M. Neamtu, and L. L. Schumaker,
BernsteinBezier polynomials on spheres and spherelike surfaces,
Comput. Aided Geom. Design. 13 (1996), 333349.
Abstract:
In this paper we discuss a natural way to define barycentric
coordinates on the sphere and on general spherelike surfaces.
This leads to a theory of BernsteinBezier polynomials which
parallels the familiar planar case. Our constructions are
based on a study of homogeneous polynomials on trihedra in
$\RR^3$. The special case of BernsteinBezier polynomials on
a circle is considered in detail.

P. Alfeld, M. Neamtu, and L. L. Schumaker,
Circular
BernsteinBezier polynomials, Mathematical Methods in CAGD,
M. Daehlen, T. Lyche, and L. L. Schumaker (eds), Vanderbilt
University Press, 1995, 1120.
Abstract:
We discuss a natural way to define barycentric
coordinates associated with circular arcs.
This leads to a theory of BernsteinBezier polynomials
which parallels the familiar interval case,
and which has close connections to trigonometric polynomials.

P. E. Koch, T. Lyche, M. Neamtu, and L. L. Schumaker,
Control curves and knot insertion for trigonometric splines,
Adv. Comp. Math. 3 (1995), 405424.
Abstract:
We introduce control curves for trigonometric splines
and show that they have properties
similar to those for classical polynomial splines.
In particular, we discuss
knotinsertion algorithms, and show that as more and more
knots are inserted into a trigonometric spline,
the associated control curves converge to the spline.
In addition, we establish a convexhull property and
a variationdiminishing result.

D. Gonsor and M. Neamtu,
Nonpolynomial polar forms,
Curves and Surfaces II (P. J. Laurent, A. Le Mehaute, and
L. L. Schumaker, Eds.), AKPeters, Wellesley, MA, 1994, 193200.
Abstract:
We begin by defining the polar form for a special type of
function, namely a trigonometric polynomial, in order to
illustrate the similarities between trigonometric polar
forms and polynomial polar forms. After deriving properties
and developing some results concerning trigonometric polar
forms, we consider the generalization to functions that
are elements of certain null spaces of constant coefficient
differential operators.

M. Neamtu and P. R. Pfluger,
Degenerate polynomial
patches of degree 4 and 5 used for geometrically smooth interpolation
in R^3,
Computer Aided Geometric Design 11 (1994), 451474.
Abstract:
The problem of interpolating scattered 3D data by a geometrically smooth
surface is considered. A completely local method is proposed, based on
employing degenerate triangular BernsteinB\'ezier patches. An analysis
of these patches is given and some numerical experiments with quartic
and quintic patches are presented.

P. R. Pfluger and M. Neamtu,
On degenerate surface patches,
Numerical Algorithms 5, J. C. Baltzer AG, Science Publishers,
1993, 569575.
Abstract:
A local construction of a $GC^1$ interpolating surface to given
scattered data in $\R^3$ can give rise to degenerate BernsteinB\'{e}zier
patches. That means the parametrization at
vertices is not regular in the sense that the length of the tangent
vector to any curve passing through a vertex is zero at that vertex.
This implies that the curvature of these curves tends to infinity
whenever one approaches a vertex. This fact seems to have not a negative
influence on the shape of the surface.

M. Neamtu,
Multivariate divided differences. I. Basic properties,
SIAM J. Numer. Anal. 29 (1992), 14351445.
Abstract:
The notion of the univariate divided differences is generalized to the
multivariate case.
This generalization is based on a pointwise evaluation of a certain multivariate
function. Several properties of the defined multivariate divided difference
functional
are derived and a link with the multivariate simplex splines is established. This
gives
a new generalization of the so called truncated power function, which is different
from
the one given in \cite{Dahmen79,Dahmen80}.

M. Neamtu, On discrete simplex splines and
subdivision, J. Approx. Th. 70 (1992), 358374.
Abstract:
Discrete analogues of multivariate simplex splines are
introduced. Their study yields a subdivision scheme for simplex splines.

M. Neamtu,
On approximation and interpolation of
convex functions, Approximation Theory, Spline Functions and
Applications, S. P. Singh (ed.), Kluwer Academic Publishers, Dordrecht,
Boston, 1992, 411418.
Abstract:
Some negative results concerning convexity preserving
approximation and interpolation of multivariate functions are
presented. We prove that the approximation based on both
interpolation and local operators cannot be convexity preserving,
provided the approximation space is (locally) finite dimensional.
In both cases we can dispense with the asssumption of the linearity of
the approximation operator and the assumption that the approximation
space is a space of piecewise polynomials.
Some consequences for the construction of shape preserving
approximations are discussed.

S. Auerbach, R. H. J. Gmelig Meyling, M. Neamtu and H. Schaeben,
Approximation and geometric modeling with simplex
Bsplines associated with irregular triangles, Computer Aided
Geometric Design 8 (1991), 6787.
Abstract:
Bivariate quadratic simplicial Bsplines defined by their
corresponding set of knots derived from a (suboptimal)
constrained Delaunay triangulation of the domain are
employed to obtain a C^1 smooth surface. The generation
of triangle vertices is adjusted to the areal
distribution of the data in the domain. We emphasize
here that the vertices of the triangles initially
define the knots of the Bsplines and do generally not
coincide with the abscissae of the data. Thus, this
approach is well suited to process scattered data. With
each vertex of a given triangle we associate two
additional points which give rise to six configurations
of five knots defining six linearly independent bivariate
quadratic Bsplines supported on the convex hull of the
corresponding five knots.

M. Neamtu, Multivariate Splines,
Dissertation, University of Twente, The Netherlands, 1991.
Abstract:
The dissertation is devoted to the study of theoretical and
practical aspects of multivariate splines and related topics
from the constructive approximation theory. In Section 1.1
the notion of polyhedral spline is introduced and a
brief survey of known results is given. Section 1.2 gives
an introduction to the topics of interpolation of
scattered data, BernsteinB\'ezier representation
over triangular partitions and to geometric continuity.
In Section 1.3 we are concerned with the topic of shape
preserving approximation.
In Chapters 26 we are dealing with multivariate simplex
splines. In particular, we establish the notion of
multivariate divided difference and a relation with multivariate
simplex splines, which is reminiscent of a well known relation
between Bsplines and univariate divided differences (Chapters
2 and 3). In Chapter 4 we derive certain recurrence
relations for simplex splines and study some related topics. In
Chapter 5 we introduce the notion of discrete simplex spline
and propose a subdivision scheme for the evaluation of
simplex splines. Similar questions are studied in connection with
Bernstein polynomials in Chapter 6.
In Chapter 7, written jointly with Dr. P. R. Pfluger from the
University of Amsterdam, we study the problem of interpolation
of data scattered in the three dimensional Euclidean space. We
employ the BernsteinB\'ezier representation for polynomials over
triangles. We propose a method for constructing a piecewise
polynomial interpolation which is local \ie such that the
interpolant is only affected locally by changes in the data. We
prove that the locality of the interpolant gives rise to degenerate
polynomial patches i.e., patches with coalescent
control points.
In Chapters 8 and 9 we study a number of problems of shape
preserving approximation. First, in Chapter 8, we derive some
theoretical results concerning convexity preserving preserving
approximation and interpolation. These results are of a negative
character. In particular, it is shown that approximation based both
on interpolation and local operators cannot be convexity preserving,
provided the approximation space is (locally) finite dimensional.
Some consequences for the construction of shape preserving
approximants are discussed. One of them is that piecewise polynomial
functions on fixed partitions of the domain in question are
not suitable for the purpose of convexity preserving approximation.
In Chapter 9, written jointly with Dr. B. Mulansky from the Technical
University of Dresden, we consider the problem of the existence
of shape preserving interpolation operators in general linear
topological spaces.

M. Neamtu, Subdividing multivariate polynomials in
BernsteinBezier form without de Casteljau algorithm, Curves and
Surfaces, P. J. Laurent, A. le Mehaute and L. L. Schumaker (eds.),
1991, 359362.
Abstract:
Some alternatives to the ``classical''
subdivision of Bernstein polynomials (i.e., based on utilizing the
wellknown de Casteljau algorithm), are sketched. Our schemes have
``asymptotically''
lower computational complexities and can be carried out such that the resulting
``control points'' take the precise values of the polynomial surface being
subdivided. For one particular approach, the so called discrete Bernstein basis
polynomials are introduced.

M. Neamtu and C. R. Traas, On computational aspects
of simplicial splines, Constr. Approx. 7 (1991), 209220.
Abstract:
Some new results on multivariate simplex Bsplines and
their practical application are presented. New recurrence
relations are derived based on [2] and [15]. Remarks on
boundary conditions are given and an example of an
application of bivariate quadratic simplex splines is
presented. The application concerns the approximation of
a surface which is constrained by a differential equation.

P. R. Pfluger and M. Neamtu, Geometrically smooth
interpolation by triangular BernsteinBezier patches with
coalescent control points, Curves and Surfaces, P. J. Laurent, A. Le
Mehaute and L. L. Schumaker (eds.), Academic Press, 1991, 363366.
Abstract:
The problem of interpolating discrete data in R^3 is considered. The
data consists of positional values and normal vectors. The objective
is to interpolate this data by a geometrically smooth (GC^1) surface. This
is acomplished by exploiting the so called degenerate triangular polynomial
patches, which admit a ``completely local'' interpolation scheme.

M. Neamtu, Multivariate divided differences II. Multivariate simplex
splines, 11pp, 1991.
Abstract:
This paper is a continuation of the first part of our paper
on multivariate divided differences. In this paper more
properties of the divided differences are derived and the
connection with multivariate simplex splines is further
investigated.

M. Neamtu, Discrete Bernstein polynomials, 9pp, 1991.
Abstract:
Discrete Bernstein polynomials over simplices are introduced.
A number of basic properties, analogous to the properties of
continuous Bernstein polynomials are established, such as
degree elevation, de Casteljau algorithm, and others.

M. Neamtu, Multivariate divided differences and Bsplines,
Approximation Theory VI, vol. 2, C. K. Chui, L. L. Schumaker, and
J. D. Ward (eds.), Academic Press, New York, 1989, 445448.
Abstract:
We give a definition of the multivariate divided difference
functional which is consistent with the corresponding
univariate notion. It is based on the pointwise evaluation
of a certain multivariate function. We also show its close
relation to the multivariate (simplex) Bsplines. Some
remarks on the Bspline evaluation are made.

M. Neamtu, On the recurrence relation for multivariate Bsplines
with respect to dimension, Memorandum no. 786, University of Twente,
The Netherlands, 14pp, 1989.
Abstract:
We are concerned with methods for computing with multivariate simplex
splines. The motivation came from a paper by Cohen, Lyche, and
Riesenfeld, where a new recurrence relation for multivariate simplex
splines has been derived. Extensions of the ideas in that paper yield
new recurrence relations for directional derivatives and inner
products of simplex splines. We discuss algorithmical and numerical
properties of the new relations and give a comparison with other
methods established earlier in the literature.

M. Neamtu, Splines with free knots in adaptive control of distributed
parameter systems, M.Sc. Thesis (in Slovak), Slovak Technical University,
180pp, 1988.

M. Neamtu, Adaptive control of heat systems, SVOC Report (in Slovak),
Slovak Technical University, 23pp, 1987.

M. Neamtu, Multivariate Bsplines and their evaluation,
Memorandum no. 598, University of Twente, The Netherlands, 30pp,
1986.
Abstract:
This paper is concerned with the evaluation of multivariate
Bsplines in a stable and efficient way for arbitrary spatial
dimension and arbitrary degree. An algorithm for evaluating
Bsplines is discussed using both the basic recurrence
relations for Bsplines and their derivatives and the simplex
method for obtaining nonnegative barycentric coordinates.
Some numerical experiments are included.
